Critical points for vector-valued functions 1. Introduction

Control and Cybernetics vol. 31 (2002) No. 3

Critical points for vector-valued funct ions by R.E. Lucchetti 1 , J.P. Revalski 2 and M. Thera 3 1

Dipartirn ento di Mat ema tica, Politecnico di Mila no, Via Bonardi 7, 20133 Milano, Italy E-mail: roblu [email protected] 2

Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str. Block 8, 1113 Sofia, Bulgaria E-mail: revalski @rnath. bas. bg 3

LACO , Departement de M at l H~rnat iques , Universite de Limoges, 123, Av. A. Thomas, 87060 Limoges, Fra nce E-mail : michel. [email protected]

Abstract: This paper contains a mountain pass theorem for coutinuous mappings, defined on a complete metric space and taking values in a real Banach space, ordered by a closed convex cone. We use the concept of critical point introduced by Degiovanni , Lucchetti a nd Ribarska, and we furni sh a vari ant of their result , allowing for a l oca li ~ation both of the critical point and of the critical value. Keywords: weak slope, critical point , P alais-Smale condition , rnounta in pass theorem, P areto optimum.

1.

Introduction

For a long time the search for minima/maxima, and more generally for critic 0.

2.1 Th e wea" slope of .f al :c E X w-ith '1-e:-;pect to Po i:; the cr E [0 , oo) s o that the·1·e ex·isl 5 > 0 u.nd a c onti'll:u ov.s 'II WJ!]J't ng tJ : B(x , 5) x [0, 6] --; X s·uch thai for evr::ry pai·r (:1:1 , L) E B(:c, 5) x [0 , 6] th e followin.!J a1·e lTue:

DEFI NITIO N

S'll]J'I"C'frt 'lt 'm of

(a) d( TJ(:c' , t) , :c') ::; t , (b) f(rJ( :c' ,t)) E f( :c')- crtPo- P. We rle-no le this

S'LL]J'I'W t 'U711

by lclru.fl (:c) .

It is easily seen tha t the wea k slope ldp0 fl (-) is a lower se1uicoutiuuous function iu X. In the sequel the se t Po is assumed to be of the fon11 Po = e + P , for some uonzero element e E P. In this case, as one could see, a u equivalent formul ation

Crit ical points for vector-va lu ed fun ct ions

547

(b') J('rJ( x', t)) E f(x')- (Jte- P . In the scalar case Y = IR, when we take P = [0 , oo) and Po := [1, oo) , this d~finitiou agrees with the definition of weak slope given by Degiova nni and Marzocchi (1994), see also Corvellec, Degiovanni and Marzocchi (1993) a nd Ioffe and Schwa rtzman (1996), a ud is simply denoted by ldfl (:t) (for more details see also Canino a ud Degiovanni , 1995). It is well-known that in this case the weak slope is a generali zatiou (fo r fuuctions whi ch are no t differeutiable) of the uorm of the d erivative. Therefore, in our lllOre general settiug the following defiuition agaiH given in Degiovanni, Lucchetti a nd Ribarska (2002) is natural: DEFINITIO N 2.2 A point x EX ·is said l.o be critica l for f with Tespecl to Po ldpofl (x) = 0.

'if

We continue by giving two more d efinitions: a subset C of Y is called invanani if C = C - P. Along the paper we will ofte n use, without special remark, the following elementa ry monotonic property of a given invari a ut set C : if e is a n element of the cone P t hen for a uy n, f3 E IR, with n < (3, we have C + ne C C + (Je. Further, a set F C Y is said to be 'l'ea chable from A C Y with Tespeci to Po if there exists t 2: 0 so that F - tPo C A. An equ ivalent assertion of the last definition is that for some t ;::: 0 we have:

F

c

n

(A+ t:y).

yEPo

If the set Po is of the special form e + P for some non zero e E P , then, if A is invariaut we have:

n

(A+ ty) =A+ ie.

yEf'o

We end this preliminary section by introducing son1 e lllore Hotation as well as a n appropriate scalar mouutaiu pass th eorem which will ue used la ter. (T, S ) will be a fixed pair of compact sets: by this we mean that T is a Hon- elllpty compact metric space and Sis a non-empty closed subset ofT . Given a con1pletc metric space ( Z, d) and a continuous 1nappi ng ·lj; : S --+ Z we denote by r( Z, 41) the following set: f(Z , ·¢) := {p: T--+ Z: pis contin uous, piS= ·¢ }. We will consid er on f( Z, ·lj;) the usual uniform metric p: for p1 , P2 E r( Z, lj;), p(pl,P2) := sup{d(p1(t) ,p2(t)) : t E T} . Then , (f(Z, ·Ij;),p) is a complete metric space. Finall y, we recall a version of a scalar mountain pass theorem , that one can find in Conti (1994). Below, as usu al, d(- , A) denotes the distance function in

548

R.E. LUCCHETTI, J.P. REVALSKI, M. THERA

THEOREM 2.1 Let (Z, d) be a complete metric space and 'ljJ: S---> Z be cmdin·uous. Suppose that g : Z ---> lR is a continuous functi on and set

c :=

inf

maxg(p(t)) .

pH(Z,,P) tET

Let L be a non-empty closed s·ubset of the set {z E Z : g(z) 2: c}, :;·uch that, fo ·r all p E f(Z,'Ijl) we have: p(T) n L of: 0 1\ 'lji (S) n L = 0. Then, joT any c > 0 th eTe exists Z 0 E Z with the f ollowing pToperties: 1. d( zc, L) ~ 2c ; 2. c - (1 / 2) c 2 ~ g( zc) ~ c + 2c 2 ; 3. [dg[( zc) ~ 3c. We mention that the classical mountain pass theorem takes as the set T the interval [0 , 1] a nd as S its endpoints 0 a nd 1. In this case f(Z, ·lj;) is the set of all continuous paths in Z connecting zo := ·lj;(O) wit h z 1 := '1/J(l). On the other hand , considering the above more general sett ing allows getting theoren1s as the saddle point theorem, see Rabinowitz (1986). This is the reason why we shall make use here of a 1ore general formulation, though we do not insist in furnishing a vector versio of t he saddl e point theorem.

3.

A mountain p a ss theorem

In this section we will establish a rnountaiu pass theorem for vector-valued fuu ctions. Here we make a less gen eral assumptiou on the set Po than in Degiovanui , Lucchetti and Ribarska (2002) , but ou the other !taud we get a lllOre precise lo calization both of the criti cal poin t :c and of its value. Moreover , we do not use a deforma tion lemma as it was done in Degiovauui , Lucchetti a nd Ribarska (2002), but we appeal to a scalar mountain pass theorem, based on the use of the Ekeland variational principle. Before establishing our theorem, we need to introduce some wore notions. Let us, as above, be given a complete metric space (X , d), a real B a nach space Y, a closed convex cone P in Y, and the set Po = e + P, for some uonzero element e E P. Furthermore, let us be given a continuous fun ction f : X ---> Y. We provide now a P alais-Smale cond ition s uitable for our vector-valued settiug. To introduce such a condition, let C be an invaria nt closed subset of Y, L be a non-empty closed subset of X and c be a real number. DEFINITION 3.1 A sequence {xn} C X is said to be a P alais-Srnale sequence

joT f (with Tespect to L, C and c), denoted by (PS)Lc,c-sequence, 1. [dp0 f [( xn) ---> 0; 2. d( x n, L)---> 0;

if

Critical point.s for vector-valued funct.ions

549

The j1mction f satisfies the (PShc,c. -condition if every (PShc,c-seq·uence hal! a duster point. From condition 1, and the fact that the weak slope is lower sernicontinuous, every cluster point of a (PS)Lc,c-sequence is a critical point for f. Moreover, since C is closed and invariant we have

n

(C + (c + c)e) \ (Int C + (c- c)e) = C \ Int C + ce,

e>O

and consequently every cluster point x of a (PS)Lc,c-sequence, is also such that f(x) E (C \ lnt C)+ ce. Finally, let the pair of compact sets (T, S) be given, let F be a closed invariant subset of Y such that f(X) n F =f. 0, let ·1/J: S __, f- 1 (F) be a continuous function and set r := r(f- 1 (F),·Ij;). Now, we are ready to formulate and prove our mountain pass theorem: 3.1 Let (X, d) be a complete rnetTic space, Y a real Banach space with ordering closed convex cone P and let Po = e + P, for some nonzem element e E P. Let f : X __, Y be a contin'UO'US funct-ion. S·uppose F is a closed ·invariant set and the pa·i'f' of compact sets (T, S) and the mapping 'ljJ are as above. Let C be an invariant closed convex set in Y with nonempty interior lnt C. S·uppose, mor-eover, the following conditions are tr'Ue: 1. F n f(X) is reachable fmm Int C and r =j:. 0; 2. TheTe ·is a E ~ s·uch that Vp E f :It E T: f(p(t)) rt lnt C + ae; 3. Set c := sup{ a E ~ : a fulfills condition 2} and s·uppose there is a closed S'UUSet L of X' sv.ch that, joT all p E r' f(L)n(IntC+ce)=0 1\ p(T)nL=j:.0 1\ ·lj;(S)nL=0; 4- f satisfies the (PS)Lc,c - condition. Then, f possesses a critical point x E L I!O that f (x) E ( ( C \ Int C)+ ce) n F. THEOREM

Pmof. We start with some remarks. First, the set Int C is invariant. This, together with condition 1 of the theorem, shows that the number c in 3 above is well-defined and finite. Next, the following monotonic property holds: (C + o:e) n f(X) n F c lnt C + {3e,

if {3 > o:. Let us prove it. Let {3 > o: and take any pointy E ( C +ne) n f(X) n F. Then y-ae E C and since f(X)nF is reachable from lnt C we have y-te E lnt C for some t ?': 0. We may think that t > {3. Then the point y - {3e is on the segment [y- te, y- ne[. Since C is convex, this entails y- {3e E Int C. We divide now the proof into several steps.

Step 1. Define the function g: X__, [- oo, oo]:

g(:r:)

:=

inf{n E

~:

j(x) E (C + ae)}, x EX,

550

R.E. LuCCHETTI , J.P. REVALSK I, M. TH ERA

We shall consider the restriction of the function g to the set f- 1 (F). Since the set F n f(X) is reachable from Int C, for every x E f- 1 (F), there exists at least one o: satisfying the property in th e brackets. Now, we claim that, for no y E Y, C can contain the line y + Ae, A E R Suppose the co ntrary, i.e. the existence of y E Y such that y + Ae E C , for all ). E !It Take a n elerneut x E f - 1 (F) such that f( x) rf. Int C + ae, for some a E JR: the existence of such an element is guaranteed by assumption 2. As F n f (X) is reachable from lnt C, there is t 2: 0 such that f( :c)- te E Int C. By convexity of Int C ami the fact that y + Ae E C for all A E IR, it follows t hat f (:c) - te + Ae E Iut C , for all A E R But this implies f( x) E Int C + ae: a co ntradiction. Thus, g( x ) > - oo and finally g is real valued on f- 1 (F ). Observe also that, for x E f- 1 (F) , clearly the infimum in the definition of g is attained. We shall get the existence of a critical point for f by finding a critical point of g, restricted to the set f - 1 (F), with the help of the scalar mount ain pass theorem (Theorem 2.1).

Step 2. The function g: f- 1 (F) _, lR is continuous in the induced topology on f- 1 (F). Let xo E

r

1

(F) a nd

f( xo) E (Int C

E

> 0 be arbitrary. As f( :co)

E

(C + g(:co)e) n F , then

+ (g(xo) + c)e) n F.

By the co ntinuity off there is some open subset U1 of X coutaining :c 0 so that

.f(x) C (Int C + (g( xo) + c)e)

n F 'V:c

E U1

n .r- 1 (F).

Then

g(:c) ~ g( xo) + E 'V:c

E

ul n

r 1(F).

(1)

Further, by the definition of g and the rnonotonicity propert ies of the sets C + o:e we have that f(xo) rf. (C + (g(xo) - c)e) n F. Since t he latter set is closed in Y and f is cont inuous we have the existence of a n open set U2 of X containin g x 0 and so that .f(x) rf. (C + (g(x 0 ) - c)e) n F for any :c E U2 n .f- 1 (F). Agaiu by the monotonic properties of the sets C + o:e and the definition of g we have that

g(:c) 2: g(xo)- c for each X E we see t hat

(2)

u2 n f- 1(F). Hence, upon putting u

lg(x)- g(xo) l ~

E

'Vx E

Therefore, g is contiuuou on

:=

ul n u2, by (1) and (2)

u n .r- 1 (F). f - 1 (F) .

The proof of step 2 is completed.

551

Critical point.s fo r vector-vn lu ed fun ctions

There is nothing to prove if idp0 fi(x) = 0. Then, take some 0 < a < idrof i(x) for which there exist 8 > 0 and a continuous mapping rJ : B(x , 8) x [0 , 8] ~ X so that for every (x', t) E B(:r:, 8) x [0 , 8] we have:

(a) d(TJ(x', t), x') ~ t, (b) j(TJ(x' , t)) E f(x')-ate-P. Let :r' E B(x, 8)nj - 1 (F). Then f(x') E F and since F is invariant (F-P F) we conclude that f(x') - atPo C F

for every t

=

2: 0.

Thus by (b) we get f(TJ(x', t)) E F for any x' E B(x, 8) n f - 1 (F) and t E [0 , 8]. 1 Therefore, the restriction of TJ to B(x , 8) n (F) x [0, 8] takes its values in

.r-

j - l(F).

Moreover, take any (x', t) E B(:c , 8) n .f - 1 (F) x [0 , 6]. Then

.f(x')

E

(C

+ g(:c')e) n F

whence, because C is inva riant,

f (TJ(:c', t))

+ (g(:c' )- at)e) n F.

E

(C

~

g( x') - at.

Jt follows th a t

(V) g(TJ(:c' , t))

This, together with (a), shows t hat idgi (:c) 2: a , a nd since a< lciro .f i(:c) was arbitra ry, we can therefore conclude that ici.r.ti(:c) 2: lcip0 fi(:c). The proof of the third step is completed.

Step 4. Let us now see tb at we can appl y Theorem 2.1 to the functi on y, to Z = .f- 1 (F) a nd to th e set L n 1 ( F ).

.r-

From step 2, we know that y is continuous. Further, we show t ha t c = c := infpEf maxtET g(p(t)) . Let a be so t bat for a uy p E f tb ere is [ E T witb f(p( t)) rj. Int C + ae. Then, g(p( l)) 2: a, otherwise th ere would be a < a such that .f(p([)) E C + ae and by the rnonotouic prop erties listed in the beginuiug of the proof we would have f(p(l)) E Int C + ae. The contradictiou s hows that g(p(l) ) 2: a for any p E f, and therefore, c ~ c. Suppose that > c a nd take a E lR with > a > c. By the defiu ition of c there is so me p E r such that .f(jj(t )) E C + ne for auy l E T. But this iillplies y(jj(t)) ~ a for every t E T. By th e defiuiti ou of this Jll eaus ~ a, a coutradictiou. Thus, c = 2 Fiually, as we assumed that .f(L) n (Int C + ce) = 0. we sec th at Ln (F) C {:c : g( :~;) 2: c}. Hence, we can apply Theorem 2.1 , to conclud e that , for every c > 0 tl1 ere exists :cc; E .f - 1 (F) so that:

c

c

c

c

.r-l

1. d(:&c;, L ) ~ 2c- ;

2. c - (1/2)c 2 ~ g(:c o: ) ~ c + 2c 2 ;

552

R.E. LUC C H ETTI , J.P. REVALSI\l , M. THERA

From step 3, we can conclude th a t :cE fulfills also idp0 .fi(:c" ) condition 2 amo unts to saying that

< 3c. Moreover ,

f(x y the Palais-Smale condition has a cluster p int x E L , which is critica l for f. Moreover :c satisfies the property f(x) E (C \ lnt C)+ce. Finally, siuce :rE E .f- 1 (F), the n f( :c) E F , and the proof of the theorem is completed. • REMARK. When e is au interior point of the cone P. it is easy to see that c- ae E Int C for every c E C and a > 0. T ltis impli es , in particula r , that C is reacha ble from Int C , thus a natural choice for the set F could be F = C + /,;e (k la rge). However , here it is uot ass um ed tha t C is reachable fro m Jut C. Observe also that, when the cone P lw.s iuterior points ami coHdit.ion 2 of the theorem is fulfilled, the functiou !J is everywhere fmite a ud coutiuuous. For , if e E Int P. for every :c E X aud c E C t here is a E R such tha t e- f (:~:) +c E P , so tha t /(:r) E c- P + ae C C + ae, s howiug tha t !J is finite at :c. C~utinuity follows from the argulllcut used in s tep 2 of Thcon.ut 3. 1.

4.

Pareto optima and critical points

We cud the paper with sou1e considcratious and res ults rela ted to the mountain pass theorem proved in t he previou s sectiou. Firs t , let us recall the notion of Pareto optimum, see Luc (1 989): le t Y be a real Ba11 ach space with a positive closed convex couc P which is pointed (the lat ter meaus tha t JJ, -p E P is possible only wheu p = 0). Let J :X --' Y be a llla.ppi ng frow X into Y. The poiut xo E X is said to be a lucal Pu:retu 'lll:iniuwm (resp. lllaxiiiiUiu) with resp ec t. toP if there is a ueigbourhood U of :c 0 so th a t (f (x 0 ) - P) n J( U) = {.f(:c 0 ) } (resp. (f( :co) + P) n !( V) = {f(:co)} ). In th e scala r case, it is well-knowu that if a point is a local uJiuium111 for a coutiuu ous function f , tl1en the weak slope off a t this poiut is zero. Iu otlwr words, local Ininillla a re critical points with respect to the weak s lope. Jt is the san te wheu the nwge space is a vector spa.ce as it can be easily see u by the secoml cond ition (b) in the dcfi11iti ou of t he weak slope. The situa tion witli the local Hmxin1unt , however, is di ffereut; in a general metri c one lllay have a fuuction with a local maximulll , without !taviug tlt e weak slope zero at thi s point . A s it11ple exalllple is: X:= [0 , 1], f( :1:) := :c; whe n l«f l( l ) = 1. But wh e u X is a finite clirneusiou al Ba.11ach space. tlt e answer is pos iti ve , as mentioned itt Conti a nd Lucchetti (1995) (in infiuite dil1lensional X the questiou is sti ll opeu ). Here we see that th e sau1 e result hold s when the nmge space is a Banach space.

4.1 Let Y be a flan.ach :SJJa ce unle1·ed by a clo::;ed conve:c ]Jointed cone P , lei. Po be a clu:;ed :;'abet uf P , nul contu:in'iny zem , and let f : lR"" __. Y be a cmJ,i'inuuus fand ion. S'nppu:;e :co is a local Pa1du ma.:cim:am fu.,. f 'W'tih

PROPO SITIO N

553

C ri t ica l points for vector-va lu ed function s

? ·roof. The proof goes as the aualogous result in the scalar case. We include it here for convenience of the reader. Suppose, by contradictiou , that there are CJ > 0, 8 > 0 and TJ as in the definitiou of the weak slope. We may think that 8 is so small that (f(:~; 0 ) + P) n f(B(:~; 0 , 8)) = {f(:co)}. Let(}: B(xo, 8)---> [0, %J be a continuous function such that: 1. B(:ro) > 0; 2. (}( :~;) = 0 if l x- xoll ~ 8/2. Let h : B( :~; 0 , 8) ---> B(xo , 8) be defiued as h(:c) = TJ(:c, (}(:c)). As h( :~;) = :I.: if llx- xo l = 8, it follows, by Brower theorem, that there is x such that h(:C) = x 0 . This leads to the desired contrad iction. For, 'IJ(i, (}(:t)) = :I.: o forces (}(:!;) = 0. This is readily seen by the fact that

f (xo) = f(TJ(i, (}(i)) E f (i)- CJfJ(:c)Po - P and that :I.:o is a local Pareto maximum. But this implies :T; = :I.:o (as TJ(-u, 0) = -u for a ny ·u), which coutradicts the relation (}(:c 0 ) > 0. The proof is completed . • We showed our main theorelll by usiug a n appropriate real valued function (t he function g) aimed at finding crit ical points of the given vector valued function f. A natural questiou is tlteu what kind of relations exist between the cri tical poiuts of g and f. Step 3 iu the proof shows that act ually every critical point of 9 is automatically critical for f. Can this staterneut be made more prec ise? For iustance, is a local maximum for y auto matically a local Pareto maximum for f ? This is the case for local 8t·rict rnaxillla fo r 9, as it can be easily sltowu. But in geueral the a nswer is negative, as the following example shows . EXAMPLE

4.1 Let X= IR2 , P = IR~, Po= (1, 1)

+ P, C

= (- 1, -1 ) - P and

let

T hen all th e points (:c 1 ,:r 2 ) sucl 1 t ha t rnax{:r: 1 , :r: 2 } ~ 0 are 111 axima for 9 , while only the points (:c 1 , :~; 2 ) with min{ :~; 1 , :c2} ~ 0 are Pareto maxima for .f. The points of the fo rm(0, :~; 2 ) and (:~; 1 ,0), :~; 1 ,:~; 2 < 0, are criti cal points fo r f which are not local rniuima/tuaxirn a . Ou the other haud, suppose there is a closed co nvex pointed co ne P such tl1at P \ {0} c Int P. Then , if we take as set C, C := z - P, for some z EX , in t hi s case locallll axima/miuill!a for 9 do correspon d to local vector maxima/ minima for f , as t he following propositiou shows. We do not know if a critical point for 9 of sadd le point type could again correspond to some vector minimum/ maximum for .f, or it must be something different. Pn.OPOSITION 4 .2

With the above P ,

P and C,

let

:l;

be a local Tnaxim'U'In/m-in-i-

554

R.E. LUCCHETTI, J.P. REVALSKI , M. THERA

Pmof. We prove the statement for a maximum, aud for easy notation we suppose it is a global maximum. So, suppose x is not a maximum for f. Then there is x such that f(x) E f( x) + P, and f( x) =f. f(x). Thus

f(:Y:)

E

f(x)- Int P.

f(x)

E

z-

As

P + g(x)e,

then

f( :T;) E z -

P + g(x) e -

It follows that y( x)

Int

PC lnt C + g(x) e.

•

< y( :1: ): a contradiction.

Finally, in the followi ng example we see that a Pareto maximum for f does not need to be a local maximum for g. Even more, we exhibit a point with weak slope zero for f which is Hot critical for the associated real- valued function g. EXAMPLE 4.2 Let X= ll~.2, P

= JR~,

Po= (1,1) +P, let C

f( :I:t, :c2 ) = (x1, miu{O, :c2 } + miu{O,

= (-1, -1 ) -

P ami

-xd ).

It is no t difficult to show that .f(X) = {(x 1 , x2) : x2 :S rnin{O , -:cJ}} aHd thus that (0 , 0) is a P areto maximum for f. A direct calculatioH, or reference to Proposition 4.1, s how that actually Jdp0 .fJ(O, 0) = 0. However, g does not have a critical point at (0, 0). For, it cau Le showu without rnucli effort that y(x1,:c2) = 1 + tuax{:I:1,x2} ami that JdgJ(O , O) = 1.

References CANINO, A., DEGTOVANNI, M. (1995) Nonsmootlt critical point theory a nd quasi liuear elliptic eq uations. In: Grauas, A., Frigou, M. and Sabidussi , G. eels., Topolog·icallvfethods ·in D'iffe ·renL·ial Equal·ions and !nda8ions (Montreal, PQ, 1994), 1--50, NATO Adv. Sci. lust. Ser. C Math. Phys. Sci. , 472, Kluwer Acad. Pub!., Dorclrecht. CONTI, M. ( 1994) Walking around mountains. fslil'uto LombaTdo (Rend. S'c.)

A 128, 53-70. CONTI, M. a nd LUC CHETTI, R. (1995) The Miniwax Approach to the Critical Poiut Theory. In: Lucchetti , R. aud Rcvalski , J. eels. , Recent Developmentl:i in Well-Posed Var"i ational Pmblems. Kluwer Academic Publishers, Mathematics and Its Applications, 331, 29-76. CORVELLEC , J.-N., DEGIOVANNI, M. and MARZOCCI-1!, M. (1993) Deform ation properties for conti nuous fuuctionals and critical point theory. Topol.

Methods Nonlinear Anal., 1, 151- 171. DEGIOVA NNI, M., and MARZOCCHI, M. (1994) A critical point theory for '

I ·'

- --

C rit.ical points fo r vec tor- v